2011线性代数试卷A标准答案和评分标准

Following parts of ordinary bolt connection: wall-beams, purlins, knee. Design of drainage, electrical, heating, ventilation, air conditioning, power should be subject to safety requirements for steel structures of other professions, it is forbidden to weld hanging load and increased roof and columns. Construction prepared and the construction layout first section material prepared and the processing production prepared according to Zhejiang more Palace steel limited and Wuhan intelligent elevator limited by signed of contract file and the Institute provides of steel design blueprint, I company and the Organization Technology Department, production department, procurement Department, bid do Engineering Department, related production sector for drawings triage, and according to owners requirements timely design out production and the processing detail and for has related professional sector of strictly triage, specific work process following: Bid do bid notice Technology Department add up do procurement department production department quality Department Engineering Department Technology Department production and the processing detail proposed Hou, by add up do coordination the sector of supporting work, and by figure line by varieties, and specifications, and quality for classified, and notification Procurement Department for material of market survey and procurement, by quality Department according to by purchased material of specifications varieties for quality review and test, not qualified of material varieties and specifications strongly ban for the engineering in the, After passing inspection informed the workshop of production of timber according to strict production details of production, workshop organized by the General Manager before the production technology and security sector technical safety training, safety education company to strictly three-level disclosure, training of qualified personnel is strictly prohibited to post production, production personnel must be listed to work, you must have the corresponding certificate. Road workers must have the certificate, and regular assessments should be carried out, not a skilled operator posts is strictly prohibited. Production of various items of equipment, appliances and other, non-professional is strictly prohibited actions and strict regulations, strong rewards, be eliminated. Production line set on person, title, responsibilities, and performance and safety comprehensive assessment. Material procurement process strictly handling the装订线华南农业大学期末考试试卷（A卷）2010-2011学年第2学期 考试科目：线性代数 试类型：（闭卷）考试 考试时间：120分钟学号 姓名 年级专业 题号一二三四五总分得分评阅人得分一、选择题（本大题共5小题，每小题3分，共15分）在每小题的选项中，只有一项符合要求，把所选项前的字母填在题中括号内1. 设矩阵A, B, C能进行乘法运算，那么（B）成立(A) AB = AC，A 0，则B = C(B)AB = AC，A可逆，则B = C(C)A可逆，则AB = BA(D)AB = 0，则有A = 0，或B = 02. 设A为n(n2)阶矩阵，且A2= I，其中I为单位阵（下同），则必有（C）(A)A的行列式等于1(B)A的逆矩阵等于I(C)A的秩等于n(D)A的特征值均为13设向量组线性相关，则向量组中（A）(A) 必有一个向量可以表为其余向量的线性组合(B)必有两个向量可以表为其余向量的线性组合(C) 必有三个向量可以表为其余向量的线性组合(D) 每一个向量都可以表为其余向量的线性组合4设n元齐次线性方程组的系数矩阵A的秩为r，则有非零解的充分必要条件是（B）(A) (B) (C) (D) 5. 设A为n阶方阵，是A的伴随矩阵。则：等于 (C)(A) (B) (C) (D) 得分二、填空题（本大题共5小题，每小题4分，满分20分）6. 已知行列式，则数a =3.7. 设向量组, 线性相关，则数k =.8. 设, ,则与的距离为9，内积为37.9. 设n阶实对称矩阵A的特征值分别为1, 2, , n，则使为正定矩阵的数t取值范围是10. 设矩阵A和B相似，其中A = ，B = 则0，.得分1.5CM三、计算题11.(满分8分) 设矩阵 ，计算.解答：= = = 计算正确2分， 2分12.（满分8分）计算行列式 D = 的值。解：将D的各列全部加到第一列，得D = (x+1+2+n) （3分）= (x+) （ 2分） = (x+)(x-1)(x-2)(x-n) （ 3分）13.(满分7分) 设 , 求 . 解：(A I)= （ 2分） , （ 4分）所以 A-1 = . （ 1分）用公式法酌情给步骤分得分1.5CM四、解答题14.(满分10分) 已知方程组有无穷多解，求a以及方程组的通解。解法一：由方程组有无穷多解，得，因此其系数行列式。即或。（3分）当时，该方程组的增广矩阵于是，方程组有无穷多解。分别求出其导出组的一个基础解系，原方程组的一个特解，故时，方程组有无穷多解，其通解为， （4分）当时增广矩阵，此时方程组无解。 （3分）解法二：首先利用初等行变换将其增广矩阵化为阶梯形。由于该方程组有无穷多解，得。因此，即。求通解的方法与解法一相同。15.（满分10分）求向量组， 的一个最大无关组，且将不属于最大无关组的向量用最大无关组线性表示出来1 所以为向量组的一个最大无关组. （6分） （2分） （2分） 16.(满分6分) A，B为4阶方阵，AB+2B=0，矩阵B的秩为2且|I+A|=|2I-A|=0。（1）求矩阵A的特征值；（2）A是否可相似对角化？说明原因。（3）求|A+3I|。解：（1）由知-1，2为的特征值。，故-2为的特征值，又的秩为2，即特征值-2有两个线性无关的特征向量，故的特征值为-1，2，-2，-2。 （2分）（建议只求出-1，2也给2分）（2）能相似对角化。因为对应于特征值-1，2各有一个特征向量，对应于特征值-2有两个线性无关的特征向量，所以有四个线性无关的特征向量，故可相似对角化。（2分）（3）的特征值为2，5，1，1。故=10。（2分）17.（满分10分）求一个正交变换，化二次型 为标准型。解：设： ，则： ， 1分首先求 A 的特征根和特征向量令：